Renormalization Group Functional Equations

Functional conjugation methods are used to analyze the global structure of various renormalization group trajectories, and to gain insight into the interplay between continuous and discrete rescaling. With minimal assumptions, the methods produce continuous flows from step-scaling {\sigma} functions, and lead to exact functional relations for the local flow {\beta} functions, whose solutions may have novel, exotic features, including multiple branches. As a result, fixed points of {\sigma} are sometimes not true fixed points under continuous changes in scale, and zeroes of {\beta} do not necessarily signal fixed points of the flow, but instead may only indicate turning points of the trajectories.Comment: A physical model with a limit cycle added as section IV, along with reference

The Schr\"oder functional equation and its relation to the invariant measures of chaotic maps

The aim of this paper is to show that the invariant measure for a class of one dimensional chaotic maps, $T(x)$, is an extended solution of the Schr\"oder functional equation, $q(T(x))=\lambda q(x)$, induced by them. Hence, we give an unified treatment of a collection of exactly solved examples worked out in the current literature. In particular, we show that these examples belongs to a class of functions introduced by Mira, (see text). Moreover, as a new example, we compute the invariant densities for a class of rational maps having the Weierstrass $\wp$ functions as an invariant one. Also, we study the relation between that equation and the well known Frobenius-Perron and Koopman's operators.Comment: 9 page

On the flow map for 2D Euler equations with unbounded vorticity

In Part I, we construct a class of examples of initial velocities for which the unique solution to the Euler equations in the plane has an associated flow map that lies in no Holder space of positive exponent for any positive time. In Part II, we explore inverse problems that arise in attempting to construct an example of an initial velocity producing an arbitrarily poor modulus of continuity of the flow map.Comment: http://iopscience.iop.org/0951-7715/24/9/013/ for published versio

Self and Microbiota-Derived Epitopes Induce CD4⁺ T Cell Anergy and Conversion into CD4⁺Foxp3⁺ Regulatory Cells

The physiological role of T cell anergy induction as a key mechanism supporting self-tolerance remains undefined, and natural antigens that induce anergy are largely unknown. In this report, we used TCR sequencing to show that the recruitment of CD4+CD44+Foxp3−CD73+FR4+ anergic (Tan) cells expands the CD4+Foxp3+ (Tregs) repertoire. Next, we report that blockade in peripherally-induced Tregs (pTregs) formation due to mutation in CNS1 region of Foxp3 or chronic exposure to a selecting self-peptide result in an accumulation of Tan cells. Finally, we show that microbial antigens from Akkermansia muciniphila commensal bacteria can induce anergy and drive conversion of naive CD4+CD44-Foxp3− T (Tn) cells to the Treg lineage. Overall, data presented here suggest that Tan induction helps the Treg repertoire to become optimally balanced to provide tolerance toward ubiquitous and microbiome-derived epitopes, improving host ability to avert systemic autoimmunity and intestinal inflammation

Wedge states in string field theory

The wedge states form an important subalgebra in the string field theory. We review and further investigate their various properties. We find in particular a novel expression for the wedge states, which allows to understand their star products purely algebraically. The method allows also for treating the matter and ghost sectors separately. It turns out, that wedge states with different matter and ghost parts violate the associativity of the algebra. We introduce and study also wedge states with insertions of local operators and show how they are useful for obtaining exact results about convergence of level truncation calculations. These results help to clarify the issue of anomalies related to the identity and some exterior derivations in the string field algebra.Comment: 40 pages, 9 figures, v3: section 3.3 rewritten, few other corrections, set in JHEP styl

On Kedlaya type inequalities for weighted means

In 2016 we proved that for every symmetric, repetition invariant and Jensen concave mean $\mathscr{M}$ the Kedlaya-type inequality $$\mathscr{A}\big(x_1,\mathscr{M}(x_1,x_2),\ldots,\mathscr{M}(x_1,\ldots,x_n)\big)\le \mathscr{M} \big(x_1, \mathscr{A}(x_1,x_2),\ldots,\mathscr{A}(x_1,\ldots,x_n)\big)$$ holds for an arbitrary $(x_n)$ ($\mathscr{A}$ stands for the arithmetic mean). We are going to prove the weighted counterpart of this inequality. More precisely, if $(x_n)$ is a vector with corresponding (non-normalized) weights $(\lambda_n)$ and $\mathscr{M}_{i=1}^n(x_i,\lambda_i)$ denotes the weighted mean then, under analogous conditions on $\mathscr{M}$, the inequality $$\mathscr{A}_{i=1}^n \big(\mathscr{M}_{j=1}^i (x_j,\lambda_j),\:\lambda_i\big) \le \mathscr{M}_{i=1}^n \big(\mathscr{A}_{j=1}^i (x_j,\lambda_j),\:\lambda_i\big)$$ holds for every $(x_n)$ and $(\lambda_n)$ such that the sequence $(\frac{\lambda_k}{\lambda_1+\cdots+\lambda_k})$ is decreasing.Comment: J. Inequal. Appl. (2018

Abel's Functional Equation and Eigenvalues of Composition Operators on Spaces of Real Analytic Functions

We obtain full description of eigenvalues and eigenvectors of composition operators Cϕ : A (R) → A (R) for a real analytic self map ϕ : R → R as well as an isomorphic description of corresponding eigenspaces. We completely characterize those ϕ for which Abel’s equation f ◦ ϕ = f + 1 has a real analytic solution on the real line. We find cases when the operator Cϕ has roots using a constructed embedding of ϕ into the so-called real analytic iteration semigroups.(1) The research of the authors was partially supported by MEC and FEDER Project MTM2010-15200 and MTM2013-43540-P and the work of Bonet also by GV Project Prometeo II/2013/013. The research of Domanski was supported by National Center of Science, Poland, Grant No. NN201 605340. (2) The authors are very indebted to K. Pawalowski (Poznan) for providing us with references [26,27,47] and also explaining some topological arguments of [10]. The authors are also thankful to M. Langenbruch (Oldenburg) for providing a copy of [29].Bonet Solves, JA.; Domanski, P. (2015). Abel's Functional Equation and Eigenvalues of Composition Operators on Spaces of Real Analytic Functions. Integral Equations and Operator Theory. 81(4):455-482. https://doi.org/10.1007/s00020-014-2175-4S455482814Abel, N.H.: Determination d’une function au moyen d’une equation qui ne contient qu’une seule variable. In: Oeuvres Complètes, vol. II, pp. 246-248. Christiania (1881)Baker I.N.: Zusammensetzung ganzer Funktionen. Math. Z. 69, 121–163 (1958)Baker I.N.: Permutable power series and regular iteration. J. Aust. Math. Soc. 2, 265–294 (1961)Baker I.N.: Permutable entire functions. Math. Z. 79, 243–249 (1962)Baker I.N.: Fractional iteration near a fixpoint of multiplier 1. J. Aust. Math. Soc. 4, 143–148 (1964)Baker I.N.: Non-embeddable functions with a fixpoint of multiplier 1. Math. Z. 99, 337–384 (1967)Baker I.N.: On a class of nonembeddable entire functions. J. Ramanujan Math. 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Learning near-optimal policies with Bellman-residual minimization based fitted policy iteration and a single sample path

We consider the problem of finding a near-optimal policy in continuous space, discounted Markovian Decision Problems given the trajectory of some behaviour policy. We study the policy iteration algorithm where in successive iterations the action-value functions of the intermediate policies are obtained by picking a function from some fixed function set (chosen by the user) that minimizes an unbiased finite-sample approximation to a novel loss function that upper-bounds the unmodified Bellman-residual criterion. The main result is a finite-sample, high-probability bound on the performance of the resulting policy that depends on the mixing rate of the trajectory, the capacity of the function set as measured by a novel capacity concept that we call the VC-crossing dimension, the approximation power of the function set and the discounted-average concentrability of the future-state distribution. To the best of our knowledge this is the first theoretical reinforcement learning result for off-policy control learning over continuous state-spaces using a single trajectory

Algebraic conditions for additive functions over the reals and over finite fields

Let $C$ be an affine plane curve. We consider additive functions $f: K\rightarrow K$ for which $f(x)f(y)=0$, whenever $(x,y)\in C$. We show that if $K=\mathbb{R}$ and $C$ is the hyperbola with defining equation $xy=1$, then there exist nonzero additive functions with this property. Moreover, we show that such a nonzero $f$ exists for a field $K$ if and only if $K$ is transcendental over $\mathbb{Q}$ or over $\mathbb{F}_p$, the finite field with $p$ elements. We also consider the general question when $K$ is a finite field. We show that if the degree of the curve $C$ is large enough compared to the characteristic of $K$, then $f$ must be identically zero.Comment: 11 page